Research · Papers · The quadratic hull and its defects · MF-011
On the zero sets of products of affine forms on F₂³
For affine forms ℓ_1,...,ℓ_m: 𝔽₂³ → 𝔽₂ and P = ∏_{j=1}^m ℓ_j, |Z(P)| ∈ {0,4,6,7,8}
Published 2026-08-29
For everyone
Plain summary
MF-011 settles a narrow question about a common way of building rules whose inputs and outputs are 0 or 1. The input space is F₂³, meaning the eight possible states formed from three bits. An affine form is a parity-like expression in those bits with a fixed offset. The proposed mechanism multiplies several such forms and asks where the product becomes zero. The exact result says its zero set, the states where the value is zero, can have only 0, 4, 6, 7, or 8 states. It can never have exactly two states, including {0,1}. This proves the pure state-affine-product explanation for the state-{0,1} wall in the zero-slack case, where no extra slack is available to the mechanism. The result explains the p7 state-0/1 failure diagnostic and indicates which encodings, meaning choices of state labels, or lifted constructions could evade it. The theorem applies only when the exported highest-degree coefficient has been shown to be exactly one such pure product. The register records no prior-art position.
Result
Let \(\ell_1,\ldots,\ell_m:\mathbb F_2^3\to\mathbb F_2\) be affine forms, and let
\[ Z(f)=\{x\in\mathbb F_2^3:f(x)=0\}. \]
For the product \(P=\prod_{j=1}^{m}\ell_j\), the exact zero-set theorem is
\[ |Z(P)|\in\{0,4,6,7,8\}. \]
In particular, \(P\) cannot vanish on exactly two states, so \(Z(P)\ne\{0,1\}\). The factor zero sets have sizes 0, 4, or 8, and the product zero set is a union of the corresponding hyperplanes. A top coefficient that is required to vanish exactly on \(\{0,1\}\) is therefore unreachable at zero slack through this pure state-affine-product mechanism.
The statement proves the obstruction for that mechanism. Sums, catalysts, and relational lifts remain outside the theorem.
Setting and definitions
The state space is \(\mathbb F_2^3\), with arithmetic over the two-element field. An affine form is a linear form in the three state coordinates plus a constant term. Its zero set is the collection of states on which it evaluates to zero. A product of affine forms is pure when the exported expression contains that product as the mechanism under examination, without a sum of such products or an additional relational construction.
The state-{0,1} wall is the named failure pattern in which the required coefficient has to be zero on the two states \(\{0,1\}\). Zero slack denotes the regime recorded for this wall. The theorem concerns the exported top coefficient, meaning the coefficient at the highest degree relevant to the construction, only when that coefficient is proved to have the pure product form above.
Method
The result was established by independent exact zero-set proofs from batches A and B. The register records the argument in BRIEF §B.1 and names two replay receipts: CONT batch_a_replay_receipt.json and CONT batch_b_replay_receipt.json.
The evidence supports the full cardinality statement for products of any number of affine forms. The two independent sources also support the application to a coefficient required to vanish on exactly the two named states. No broader mechanism is included in the evidence for this entry.
Discussion
MF-011 was upgraded from CONJECTURE to PROVED on 2026-08-12. Its scope is a precise structural obstruction in Boolean function analysis. The proof rules out a product of affine state forms whenever the required zero set has size two. It explains the p7 state-0/1 failure diagnostic under the additional condition that the exported top coefficient has been proved to be one pure state-affine product.
That condition carries the main limit. A sum, a catalyst coordinate, or a relational lift can change the form of the exported coefficient. Those constructions remain outside the theorem, so the result does not close every possible route around the wall. It predicts which codes or lifts may evade the pure-product obstruction, while each such candidate still requires its own analysis.
No correction, retraction, restoration, or scope flag for MF-011 appears in the INDEX corrections section, and the curation note records Corrections: NONE. The register records no prior-art position.
For everyone — the takeaway
What this means
MF-011 says a very specific two-state pattern cannot come from this particular multiplication recipe. Multiplying parity-like ingredients creates a zero wherever any ingredient is zero. On three bits, the possible sizes of that zero region skip two. That missing size is the obstruction. The result gives a clean diagnosis for the state-{0,1} wall and closes the pure-product route when its required coefficient has that form. Other construction styles remain available for study. A sum of products, an added helper coordinate, or a relation carrying extra information could require a separate result. The register places MF-011 among the proved findings and records independent exact proofs.
Register references
MF-011; BRIEF §B.1; CONT batch_a_replay_receipt.json; CONT batch_b_replay_receipt.json; prior art: the register does not record this.
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 2 of 2 receipt files bundled (7 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.